Trigonometry flashcards

Topic 6: Trigonometry

Pythagoras' theorem

In a right-angled triangle with hypotenuse c: a² + b² = c². Add the squares to find the hypotenuse; subtract to find a shorter side.

sin θ, cos θ and tan θ in a right-angled triangle

sin θ = opposite/hypotenuse cos θ = adjacent/hypotenuse tan θ = opposite/adjacent

Angle of elevation and angle of depression

Both are measured from the horizontal: elevation upwards to the line of sight, depression downwards. The depression from A to B equals the elevation from B to A (alternate angles).

The sine rule

a/sin A = b/sin B = c/sin C (side a is opposite angle A). Use it when you know a side and its opposite angle.

The cosine rule (finding a side and finding an angle)

a² = b² + c² − 2bc cos A cos A = (b² + c² − a²)/(2bc) Use it with two sides and the included angle, or with all three sides.

Area of a triangle using sine

Area = ½ab sin C, where C is the angle between sides a and b.

Two sides are 5 cm and 8 cm with an included angle of 60°. Find the third side.

a² = 25 + 64 − 2 × 5 × 8 × cos 60° = 89 − 40 = 49 a = 7 cm

sin(180° − x) and cos(180° − x)

sin(180° − x) = sin x, so sin 140° = sin 40°. cos(180° − x) = −cos x, so a negative cosine means an obtuse angle.

The ambiguous case of the sine rule

Given two sides and a non-included angle, sin B can give B or 180° − B. Keep the obtuse value only if the angles still add up to less than 180°.

Space diagonal of a cuboid l × w × h

√(l² + w² + h²) Example: 4 × 3 × 12 cm gives √(16 + 9 + 144) = 13 cm.

Angle between a line and a plane

The angle between the line and its projection on the plane. Drop a perpendicular from the top of the line to the plane, draw the right-angled triangle, then use a trig ratio.

4024 accuracy for answers on the calculator paper

Non-exact answers to 3 significant figures and angles in degrees to 1 decimal place, unless the question says otherwise. Do not round until the final answer.